\(\int \frac {\text {arccosh}(a x)^2}{(c-a^2 c x^2)^{7/2}} \, dx\) [224]

Optimal result

Integrand size = 22, antiderivative size = 429 \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {x}{3 c^3 \sqrt {c-a^2 c x^2}}-\frac {x}{30 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arccosh}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arccosh}(a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {16 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}} \]

[Out]

1/5*x*arccosh(a*x)^2/c/(-a^2*c*x^2+c)^(5/2)+4/15*x*arccosh(a*x)^2/c^2/(-a^2*c*x^2+c)^(3/2)-1/3*x/c^3/(-a^2*c*x
^2+c)^(1/2)-1/30*x/c^3/(-a*x+1)/(a*x+1)/(-a^2*c*x^2+c)^(1/2)+8/15*x*arccosh(a*x)^2/c^3/(-a^2*c*x^2+c)^(1/2)+1/
10*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/c^3/(-a^2*x^2+1)^2/(-a^2*c*x^2+c)^(1/2)+4/15*arccosh(a*x)*(a*x-1
)^(1/2)*(a*x+1)^(1/2)/a/c^3/(-a^2*x^2+1)/(-a^2*c*x^2+c)^(1/2)+8/15*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/
a/c^3/(-a^2*c*x^2+c)^(1/2)-16/15*arccosh(a*x)*ln(1-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*(a*x-1)^(1/2)*(a*x+1)^
(1/2)/a/c^3/(-a^2*c*x^2+c)^(1/2)-8/15*polylog(2,(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*(a*x-1)^(1/2)*(a*x+1)^(1/
2)/a/c^3/(-a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5901, 5899, 5913, 3797, 2221, 2317, 2438, 5912, 5914, 39, 40} \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {8 \sqrt {a x-1} \sqrt {a x+1} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {8 x \text {arccosh}(a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {8 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}-\frac {16 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x) \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {4 x \text {arccosh}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {x \text {arccosh}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {x}{3 c^3 \sqrt {c-a^2 c x^2}}-\frac {x}{30 c^3 (1-a x) (a x+1) \sqrt {c-a^2 c x^2}} \]

[In]

Int[ArcCosh[a*x]^2/(c - a^2*c*x^2)^(7/2),x]

[Out]

-1/3*x/(c^3*Sqrt[c - a^2*c*x^2]) - x/(30*c^3*(1 - a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (Sqrt[-1 + a*x]*Sqrt[1
+ a*x]*ArcCosh[a*x])/(10*a*c^3*(1 - a^2*x^2)^2*Sqrt[c - a^2*c*x^2]) + (4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh
[a*x])/(15*a*c^3*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) + (x*ArcCosh[a*x]^2)/(5*c*(c - a^2*c*x^2)^(5/2)) + (4*x*Ar
cCosh[a*x]^2)/(15*c^2*(c - a^2*c*x^2)^(3/2)) + (8*x*ArcCosh[a*x]^2)/(15*c^3*Sqrt[c - a^2*c*x^2]) + (8*Sqrt[-1
+ a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(15*a*c^3*Sqrt[c - a^2*c*x^2]) - (16*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh
[a*x]*Log[1 - E^(2*ArcCosh[a*x])])/(15*a*c^3*Sqrt[c - a^2*c*x^2]) - (8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*PolyLog[2,
E^(2*ArcCosh[a*x])])/(15*a*c^3*Sqrt[c - a^2*c*x^2])

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rule 40

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-x)*(a + b*x)^(m + 1)*((c + d*x)^(m
+ 1)/(2*a*c*(m + 1))), x] + Dist[(2*m + 3)/(2*a*c*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /;
FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5899

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcCosh
[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Dist[b*c*(n/d)*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])], Int[x
*((a + b*ArcCosh[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ
[n, 0]

Rule 5901

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(
p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 1))), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a
+ b*ArcCosh[c*x])^n, x], x] - Dist[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[x*(
1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] &&
EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 5912

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(
x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]

Rule 5913

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/e, Subst[Int[(
a + b*x)^n*Coth[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 5914

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*
(-1 + c*x)^p)], Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a,
b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {x \text {arccosh}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{5 c}-\frac {\left (2 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x \text {arccosh}(a x)}{(-1+a x)^3 (1+a x)^3} \, dx}{5 c^3 \sqrt {c-a^2 c x^2}} \\ & = \frac {x \text {arccosh}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arccosh}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{15 c^2}-\frac {\left (2 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x \text {arccosh}(a x)}{\left (-1+a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (8 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x \text {arccosh}(a x)}{(-1+a x)^2 (1+a x)^2} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}} \\ & = \frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arccosh}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arccosh}(a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {1}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{10 c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (8 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x \text {arccosh}(a x)}{\left (-1+a^2 x^2\right )^2} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (16 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x \text {arccosh}(a x)}{1-a^2 x^2} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}} \\ & = -\frac {x}{30 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arccosh}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arccosh}(a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {1}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (4 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {1}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {\left (16 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}(\int x \coth (x) \, dx,x,\text {arccosh}(a x))}{15 a c^3 \sqrt {c-a^2 c x^2}} \\ & = -\frac {x}{3 c^3 \sqrt {c-a^2 c x^2}}-\frac {x}{30 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arccosh}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arccosh}(a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (32 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\text {arccosh}(a x)\right )}{15 a c^3 \sqrt {c-a^2 c x^2}} \\ & = -\frac {x}{3 c^3 \sqrt {c-a^2 c x^2}}-\frac {x}{30 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arccosh}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arccosh}(a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {16 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (16 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{15 a c^3 \sqrt {c-a^2 c x^2}} \\ & = -\frac {x}{3 c^3 \sqrt {c-a^2 c x^2}}-\frac {x}{30 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arccosh}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arccosh}(a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {16 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (8 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arccosh}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}} \\ & = -\frac {x}{3 c^3 \sqrt {c-a^2 c x^2}}-\frac {x}{30 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arccosh}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arccosh}(a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {16 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 1.07 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.51 \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {a x \left (10+\frac {1}{1-a^2 x^2}\right )+2 \left (8 \sqrt {\frac {-1+a x}{1+a x}}+a x \left (-8+8 \sqrt {\frac {-1+a x}{1+a x}}-\frac {3}{\left (-1+a^2 x^2\right )^2}+\frac {4}{-1+a^2 x^2}\right )\right ) \text {arccosh}(a x)^2+\frac {\left (\frac {-1+a x}{1+a x}\right )^{3/2} \text {arccosh}(a x) \left (-11+8 a^2 x^2+32 \left (-1+a^2 x^2\right )^2 \log \left (1-e^{-2 \text {arccosh}(a x)}\right )\right )}{(-1+a x)^3}-16 \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \operatorname {PolyLog}\left (2,e^{-2 \text {arccosh}(a x)}\right )}{30 a c^3 \sqrt {c-a^2 c x^2}} \]

[In]

Integrate[ArcCosh[a*x]^2/(c - a^2*c*x^2)^(7/2),x]

[Out]

-1/30*(a*x*(10 + (1 - a^2*x^2)^(-1)) + 2*(8*Sqrt[(-1 + a*x)/(1 + a*x)] + a*x*(-8 + 8*Sqrt[(-1 + a*x)/(1 + a*x)
] - 3/(-1 + a^2*x^2)^2 + 4/(-1 + a^2*x^2)))*ArcCosh[a*x]^2 + (((-1 + a*x)/(1 + a*x))^(3/2)*ArcCosh[a*x]*(-11 +
8*a^2*x^2 + 32*(-1 + a^2*x^2)^2*Log[1 - E^(-2*ArcCosh[a*x])]))/(-1 + a*x)^3 - 16*Sqrt[(-1 + a*x)/(1 + a*x)]*(
1 + a*x)*PolyLog[2, E^(-2*ArcCosh[a*x])])/(a*c^3*Sqrt[c - a^2*c*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(793\) vs. \(2(395)=790\).

Time = 1.23 (sec) , antiderivative size = 794, normalized size of antiderivative = 1.85

method	result	size
default	\(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 a^{5} x^{5}-20 a^{3} x^{3}-8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+15 a x +16 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-8 \sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (-64 \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{7} x^{7}-64 \,\operatorname {arccosh}\left (a x \right ) a^{8} x^{8}-32 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{7} x^{7}-32 a^{8} x^{8}+248 \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{5} x^{5}+280 \,\operatorname {arccosh}\left (a x \right ) a^{6} x^{6}+126 x^{5} a^{5} \sqrt {a x -1}\, \sqrt {a x +1}+142 a^{6} x^{6}+80 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )^{2}-340 a^{3} x^{3} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-456 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )-156 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}-265 a^{4} x^{4}-190 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2}+165 a x \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+328 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )+62 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +235 a^{2} x^{2}+128 \operatorname {arccosh}\left (a x \right )^{2}-88 \,\operatorname {arccosh}\left (a x \right )-80\right )}{30 \left (40 a^{10} x^{10}-215 a^{8} x^{8}+469 a^{6} x^{6}-517 a^{4} x^{4}+287 a^{2} x^{2}-64\right ) c^{4} a}-\frac {16 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right )^{2}}{15 c^{4} a \left (a^{2} x^{2}-1\right )}+\frac {16 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{15 c^{4} a \left (a^{2} x^{2}-1\right )}+\frac {16 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{15 c^{4} a \left (a^{2} x^{2}-1\right )}+\frac {16 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{15 c^{4} a \left (a^{2} x^{2}-1\right )}+\frac {16 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{15 c^{4} a \left (a^{2} x^{2}-1\right )}\)	\(794\)

[In]

int(arccosh(a*x)^2/(-a^2*c*x^2+c)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/30*(-c*(a^2*x^2-1))^(1/2)*(8*a^5*x^5-20*a^3*x^3-8*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^4*x^4+15*a*x+16*a^2*x^2*(a*
x-1)^(1/2)*(a*x+1)^(1/2)-8*(a*x-1)^(1/2)*(a*x+1)^(1/2))*(-64*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^7*x^7-
64*arccosh(a*x)*a^8*x^8-32*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a^7*x^7-32*a^8*x^8+248*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+
1)^(1/2)*a^5*x^5+280*arccosh(a*x)*a^6*x^6+126*x^5*a^5*(a*x-1)^(1/2)*(a*x+1)^(1/2)+142*a^6*x^6+80*a^4*x^4*arcco
sh(a*x)^2-340*a^3*x^3*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)-456*a^4*x^4*arccosh(a*x)-156*a^3*x^3*(a*x-1)^(1
/2)*(a*x+1)^(1/2)-265*a^4*x^4-190*a^2*x^2*arccosh(a*x)^2+165*a*x*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+328*
a^2*x^2*arccosh(a*x)+62*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x+235*a^2*x^2+128*arccosh(a*x)^2-88*arccosh(a*x)-80)/(40
*a^10*x^10-215*a^8*x^8+469*a^6*x^6-517*a^4*x^4+287*a^2*x^2-64)/c^4/a-16/15*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^
2*x^2-1))^(1/2)/c^4/a/(a^2*x^2-1)*arccosh(a*x)^2+16/15*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^4/
a/(a^2*x^2-1)*arccosh(a*x)*ln(1+a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+16/15*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^
2-1))^(1/2)/c^4/a/(a^2*x^2-1)*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))+16/15*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-
c*(a^2*x^2-1))^(1/2)/c^4/a/(a^2*x^2-1)*arccosh(a*x)*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))+16/15*(a*x+1)^(1/2)*
(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^4/a/(a^2*x^2-1)*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))

Fricas [F]

\[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(arccosh(a*x)^2/(-a^2*c*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*c*x^2 + c)*arccosh(a*x)^2/(a^8*c^4*x^8 - 4*a^6*c^4*x^6 + 6*a^4*c^4*x^4 - 4*a^2*c^4*x^2 + c^
4), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate(acosh(a*x)**2/(-a**2*c*x**2+c)**(7/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(arccosh(a*x)^2/(-a^2*c*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

integrate(arccosh(a*x)^2/(-a^2*c*x^2 + c)^(7/2), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(arccosh(a*x)^2/(-a^2*c*x^2+c)^(7/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{{\left (c-a^2\,c\,x^2\right )}^{7/2}} \,d x \]

[In]

int(acosh(a*x)^2/(c - a^2*c*x^2)^(7/2),x)

[Out]

int(acosh(a*x)^2/(c - a^2*c*x^2)^(7/2), x)